D EFINITENESS AS M AXIMAL I NFORMATIVENESS⇤
K AI VON F INTEL , DANNY F OX , AND S ABINE I ATRIDOU
Massachusetts Institute of Technology
Abstract
We argue that definites are interpreted as denoting the maximally informative object that falls
under the relevant predicate.
Given Irene Heim’s well-known contributions to the domain of (in)definiteness as well as that of
presuppositions, we hope to show our respect for her, as well as our gratefulness for everything
she has taught us, with a few thoughts on these topics.
1 An influential analysis
This very short paper is concerned with the way the definite article the works. Consider the
two definite phrases in (1). The singular one in (1a) presupposes that there is one girl in the
relevant domain (existential presupposition) and that there is no more than one girl (uniqueness
presupposition). The phrase refers to the unique girl. The plural definite in (1b), by contrast,
presupposes existence but not uniqueness. The phrase refers to the entire collection of girls.
(1) a. the girl
b. the girls
There are two influential proposals on the meaning of the definite article that try to unify the
singular and plural occurrences: Sharvy (1980) and Link (1983). Regardless of their differences,
they share the insight that the definite article the presupposes that the extension of its complement
has a unique maximal member based on an ordering of the elements:
⇤ This short paper is a collaborative work long in the making. Ideas towards it are found in von Fintel and Iatridou
(2005) and there is a short report on this work in Fox and Hackl (2006, pp.548f). There have also been reports by
Schlenker (2012) and Erlewine and Gould (2014). We are grateful to Luka Crnič and an anonymous reviewer for very
helpful comments, not all of which we were able to do full justice to in the current state of the project.
© 2014 Kai von Fintel, Danny Fox, and Sabine Iatridou. In: Luka Crnič and Uli Sauerland (eds.),
The Art and Craft of Semantics: A Festschrift for Irene Heim, vol. 1, MITWPL 70, pp. 165–174
166
Kai von Fintel, Danny Fox, and Sabine Iatridou
(2) Jthe f K (where f is of type ha,ti) is defined only if there is a unique maximal object x st.
f (x) is true (based on an ordering on elements of type a). The reference of the f (when
defined) is this unique maximal element.
Let’s unpack this a bit. A “maximal object” is an object that, with respect to a certain ordering
relation, has no other object ordered higher than it. Note that unique existence of a maximal object
is not in general entailed by the existence of an ordering. It could be that there are two maximal
objects, if there are two objects that have nothing ordered above them. But the proposal in (2),
encodes as part of the meaning of the definite article that there is a unique maximal object.
The idea in applying this to the cases in (1) is that the ordering involved is based on the
part/whole relationship between individuals. Both the singular and the plural NPs in (1) have
denotations in the domain of individuals. Singular NPs are only true of atomic individuals, that
is individuals which either do not have parts, or if they do have parts, those parts are not in the
denotation of the noun. For example, while the denotation of the NP girl might have parts, none
of the parts can be characterized as girl.1
Now, if there were two atomic individuals that are girls, neither would be part of the other,
so they would both be maximal. So, the only time there will be a uniquely maximal atomic
girl is when there is one girl (the existential presupposition) and only one girl (the uniqueness
presupposition).
The denotation of a plural NP results from the closure of the singular NP under sum formation.
Again, the domain has to have individuals in it (the existential presupposition). Let us say that it
consists of three girls, a, b, c. The denotation of the plural NP girls consists of the atomic girls
a, b, c and any sum of two or more of them: a + b, b + c, a + c, a + b + c. The maximal object with
respect to the part/whole ordering is a + b + c and there is no other maximal object. Applying the
definite article to a plural NP then, yields the uniquely maximal object, a + b + c.
By the above rationale, we have a uniqueness presupposition in the case of the singular NP, but
not in the case of the plural.
2 Our Alternative
We argue for a different meaning for the definite article:
(3) a. Jthe f K is defined in w only if there is a uniquely maximal object x, based on the ordering
f , such that f (w)(x) is true. The reference of the f (when defined) is this maximal
element.
b. For all x, y of type a and property f of type hs, ha,tii, x f y iff l w.f (w)(x) entails
l w.f (w)(y).
Basically (3a,b) only differs from (2) in the nature of the criterion that imposes the ordering.
The ordering in (2) looks at the individuals the predicate f is true of and at their part/whole
ordering. The ordering in (3a,b) looks at the same individuals but then looks at the information
contained in claiming that they are f -individuals. The criterion for the ordering is informativeness.
1 The
alternative we will propose in section 2 is committed (as far as we can see) to the idea that atomicity is
defined independently of the predicate girl. Specifically we will need it to hold for all X, Y such that X Y that the
proposition l w. X is a girl or girls in w entails the proposition l w. Y is a girl or girls in w. This follows as long as
plurality involves closure under sum-formation of elements that are inherently atomic.
Definiteness as Maximal Informativeness
167
The uniquely maximal object is the one that creates the most informative true proposition. The
most informative proposition is defined as the proposition that entails all other relevant propositions
(aka as the “strongest proposition”).
Since the maximal object by f is the one that yields the most informative true proposition of
the form l w.f (w)(x), we will call this object “the most informative object given f ” or sometimes
simply “the most informative object”.
3 Comparing the analyses
To compare our proposal and the Sharvy/Link proposal in (2), we will go through certain sample
cases.
The first desired result is accounting for the fact that there is a uniqueness presupposition in the
case where the definite article combines with a singular NP but not when it combines with a plural
NP. We saw above how the Sharvy/Link account derives this fact.
We predict uniqueness for singular definite descriptions in a very similar way to Sharvy and
Link: with a singular predicate we would be comparing the informativeness of claiming that atomic
individual a falls under the predicate to parallel claims about other atomic individuals. None of
these claims logically entail the others, so there can only be a maximally informative object if there
is a single one.
We predict uniqueness to disappear for plural definite descriptions such as the one in (1b), for
the same reasons as Sharvy and Link: a plural property of individuals f , like girls, which is true
of a plurality of individuals will necessarily be true of its atomic parts, since it is a distributive
property. Hence, the largest element by the ordering of informativity will be the largest element by
the part-whole ordering.
Assume that Ivy, Olivia, and Emma are the only girls. Then the girls will denote the maximal
plurality made up of Ivy+Olivia+Emma. This is predicted by (2) because this is the maximal
girl-plurality. That is, it is the object that is ordered highest by the part/whole criterion. It contains
all other objects that girls is true of.
We derive the same result on our account: the plurality made up of those three individuals
is the most informative object: it is this plurality from which we can deduce the “girlness” of
all other girls. Any smaller plurality would be less informative. For example, from the plurality
Olivia+Emma being girls we cannot deduce that Ivy is a girl.
So both the Sharvy/Link proposal and our proposal pick out the same individual in the case of
(1b). In fact, the Sharvy/Link proposal and ours make the same prediction for a number of cases:
all the properties that are upward monotone with respect to informativity. By “upward monotone”
we mean here that a property yields a more informative claim the “bigger” the object is that it is
predicated of. Conversely, “downward monotone” will mean that the “smaller” the object is, the
more informative the claim is that the property holds of it.
Properties of degrees such as l d. Miranda is d tall or l n. Miranda has n many children
are upward monotone: saying that Miranda is 1.65m is more informative than saying that she
is 1.60m.2 Saying Miranda has 4 children is more informative than saying that she has 3. On our
account, definite descriptions such as Miranda’s height or the number of children that Miranda
2 In the literature on degree semantics, the predicates that correspond to downward entailing properties are often
referred to as “downward scalar”.
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Kai von Fintel, Danny Fox, and Sabine Iatridou
has will refer (in any world w) to the maximal informative object in the extension of the property
in w. Therefore, they will pick out 1.65 and 4, respectively.
The Sharvy/Link account picks out the same objects for these definite descriptions: It will pick
out 4 and not 3 because it is more highly ordered on the relevant ordering on the set of degrees.
And for the same reason it will pick out 1.65 over 1.60.
We propose that the Sharvy/Link theory got the right results only because the focus was limited
to upward monotone properties (in the sense defined above). The different predictions between the
Sharvy/Link account and ours start showing up once we look at properties that are not upward
monotone with respect to informativity.
Consider then properties that are downward monotone with respect to informativity. These are
cases where the smallest amount/object is most informative. Here is such a case:
(4) the amount of walnuts sufficient to make a pan of baklava
Propositions of the form d-much walnuts is sufficient to make a pan of baklava become more
informative the smaller d is. We thus correctly predict that the definite description in (4) should
refer to the minimum amount of walnuts that would yield a true proposition, i.e. to the minimum
amount that would suffice for baklava baking.3
On the other hand, according to the Sharvy/Link account, the definite description in this
sentence should be undefined (and the sentence should thereby suffer form presupposition failure).
The reason is that (given the monotonicity) there can be no maximal amount of walnuts that is
sufficient to make a pan of baklava: if an amount of walnuts, f , suffices to make a pan of baklava,
so does any amount larger than f .
(These kinds of examples are familiar from Beck and Rullmann (1999), where they play a
crucial rule in thinking about maximality in questions. Beck and Rullmann argue that the relevant
notion of maximality of questions is maximal informativeness. We are obviously indebted to their
work.)
So we see that a definite description of the form the f alternates between referring to the
minimal or the maximal individual in the extension depending on the monotonicity with respect
to informativity of property f . We get a maximality effect when f is upward monotone and a
minimality effect when f is downward monotone as in (4).
Once the principle is clear, it is easy to construct further cases showing a minimality effect:
consider, for example, the number of Greek soldiers who together can destroy the Trojan army.
For Sharvy/Link, this would again result in a presupposition failure because there is no maximal
number of Greek soldiers that together can destroy the Trojan army: if a certain number of soldiers
can destroy the Trojan army, any larger number of soldiers can too. For us, on the other hand, the
description will pick out the minimal number of soldiers that together can destroy the Trojan army,
3 There are two interesting points that we won’t focus
on here. (i) Note that sufficient to make a pan of baklava does
not mean that as soon as you have that amount of walnuts you have thereby made baklava, as one might have thought
if educated in the language of logic (where “sufficient condition for p” means a condition that guarantees the truth of
p). This is a general feature of the use of sufficient in natural language. (ii) Note that one could use necessary in (4)
and convey the very same meaning. This otherwise puzzling fact (necessary and sufficient surely don’t mean the same
thing) follows from our proposal. We can refer here to relevant discussion in von Fintel and Iatridou (2007).
Definiteness as Maximal Informativeness
169
because that is the most informative such number: once we know that number we can deduce that
all larger numbers would also do.4
A particularly nice way to see the effects of variable direction of informativity comes from a
pair of predicates in Aloni 2007: “When told John can spend 150 euro, people normally conclude
that John cannot spend more. But when told John can live on 150 euro, they conclude that John
cannot live on less.” Using these predicates in definite descriptions thus gives rise to two different
directions of strength of informativity:
(5) a. the amount of money John can spend (on vacation)
b. the amount of money John can live on
The definite description in (5a) refers to the maximum amount of money John can spend because
that is the most informative object. The definite description in (5b) refers to the minimum amount
of money John can live on, because that is the most informative object in this case.5
Finally, let us go to properties that are non-monotone with respect to informativity. What
would it mean to be non-monotone with respect to informativity? It would mean that there are no
inferences to be drawn either way: from f (x) we can not infer anything about any other f (y). If
so, in order to have a unique maximally informative object, there would have to be just a single
object falling under f . As soon as there is more than one such object, we predict presupposition
failure. We will see that the Sharvy/Link account makes again a different prediction.
Let us set up a particular context. Imagine that you are trying to fit books (x, y, z, w, v . . .) on
shelves of various size (a, b, c . . .). Suppose that book x together with book y fit perfectly on shelf
a (that is, x and y together fill exactly the space provided by a, nothing more, nothing less), and
books x, y, and z together fit perfectly on shelf b. Suppose also that no other combination of books
fits perfectly on any other shelf. In this context, consider sentence (6).
4 This
is not entirely innocuous. It is not obvious that larger numbers can, in fact, be deduced. After all, there could
be circumstances where adding soldiers to an otherwise victorious army will (by necessity) reverse the outcome of the
war. (Assume for example that the war is lengthy and that all the soldiers need to be fed, else there would be a revolt.
Under such circumstances an efficient army with 100 soldiers might win the war, but it is conceivable that an army
with 1000 soldiers will necessarily loose.) A full account of the minimality effect, would have to demonstrate that the
property with which the definite article combines is downward monotone despite this counter-argument.
One path towards an account would be to develop a semantics under which together P holds of a plural individual
X the moment it holds of a sub-part of X. If there is a plurality Y of 100 Greek soldiers such that Y together beat the
Trojan army, it would follow, under such a semantics, that any plurality X larger than Y beat the Trojan army. The
intuition that this is not the case would then have to be attributed to something outside the basic semantics, probably
to a Scalar Implicature. Reasons to think that a Scalar Implicature might be involved can be found in the observation
that sometimes the intuition is different. (E.g. The books in Widener library together surely contain the solution to
our problems can be true even if many books do not contribute to the solution in any way. Complicated issues related
to “team credit” arise here which we haven’t thought through seriously.) However, pursuing such an account would
require quite a bit of work and in particular in the context of this paper would require that something special be said to
derive the non-monotonicity assumed in our analysis of (6) below (for example that a necessary exhaustivity operator
is present). Another path would be to modify f so that it does not make reference to logical entailment but to some
other notion of informativity. We will, unfortunately, have to leave the matter to another occasion.
5 Here too, a complete account would require a semantics for the properties that combine with the definite article,
showing that it is upward monotone in (5a) and downward monotone in (5b).
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Kai von Fintel, Danny Fox, and Sabine Iatridou
(6) #Pass me the books that together fit perfectly on a shelf!
On our account, the definite description the books that together fit perfectly on a shelf suffers
from presupposition failure. We believe that this is the actual judgment for this sentence. The
reason that our account predicts presupposition failure is that the predicate books that together fit
perfectly on a shelf is not monotone. From x + y + z fitting perfectly together on a shelf, we cannot
conclude about any other combination of books that it will fit perfectly together on a shelf.6
On the other hand, for Sharvy/Link this NP should be acceptable: it should refer to the largest
collection of books that fit perfectly on a shelf, namely, x + y + z.
To summarize then so far: in upward monotone environments, our account and the Sharvy/Link
account make the same predictions. In downward monotone environments, we correctly predict
that the definite description will pick out the minimal object/amount but Sharvy/Link predict a
presupposition failure. In non-monotone environments where the NP predicate is true of more than
one (plural) object, we correctly predict a presupposition failure, whereas Sharvy/Link predict that
the definite description should meet no problem whatsoever as long as the objects have a part/whole
structure.
4 Temporal definites
An alternation similar to the one between minimality and maximality shows up in the domain of
times as well.7 Consider the following two definite descriptions:
(7) a. January 5th 1999 is the date until which Bill will be living in Boston.
b. January 5th 1999 is the date since which Bill has been living in Boston
In (7a) the definite description the date until which Bill will be living in Boston picks out the
latest date until which Bill will be living in London.
On the other hand, in (7b) the definite description the date since which Bill has been living in
Paris picks out the first day since which Bill has been living in Paris.
For sure, this is partly the result of the meaning of until and since. That is, ‘forward looking in
time” and “backward looking in time’ is part of the lexical semantics of until and since. But what
is not part of their lexical semantics is the “latest day” in the case of until and the “earliest day” in
the case of since. We can see this in the following:
6A
reviewer points out that judgments change if the two sets of books (those that fit perfectly on shelf a and those
that perfectly on shelf b) are mutually exclusive, that is if there’s no overlap between the two sets. We tentatively
agree and think that this follows from our proposal, once we observe that the predicate together fit perfectly on a shelf
can be interpreted distributively relative to a cover (in the sense of Schwarzschild (1996)) each member of which is
non-atomic.
(i) The books in this pile (all) fit perfectly on a shelf.
We further point out that predicates that resist distributivity altogether, e.g., as a team can win the competition, yield
a clear uniqueness presupposition:
(ii) Introduce me to the athletes who as a team can win the competition.
To our ears, (ii) yields presupposition failure in a context where there are two disjoint sets of athletes each of which
can win the competition as a team.
7 This section is related to discussion in von Fintel and Iatridou (2005).
Definiteness as Maximal Informativeness
171
(8) a. For sure he will be in Boston until the wedding. Maybe he will be here after that as well.
b. For sure he has been in Boston since the wedding but maybe he was in town long before
that as well.
If the “the latest day” and the “earliest day” were part of the meaning of these items, (8a) and
(8b) would be contradictions or at any rate, infelicitous. So we conclude that “earliest” and “latest”
are not part of the meaning of the the temporal prepositions.8 Instead, the interval that is set up by
these items seems to have the same behavior as numerals, which specify a lower bound only:
(9) For sure Miranda has 3 children, but maybe she has more.
We saw earlier how we derived that the number of children that Miranda has picks out the
maximal number, as that is the most informative. So unlike (9), (10) is very odd:9
(10) #(For sure) the number of children that Miranda has is three but maybe it is more.10
The contrast beween (9) and (10) is due to the fact that Miranda having at most 3 children is
part of a scalar implicature in the first clause in (9) and this implicature can be and is cancelled by
the second clause. But in (10) it is part of the semantics, given that it is contributed by the meaning
of the definite description.
In that light, compare (8a,b) to the following, which are much worse:
(11) a. #January 5, 1999 is the date until which Bill will be living in Boston but maybe he will
stay beyond that.
b. #January 5, 1999 is the date since which Bill has been living in Boston but maybe he
was here before that.
In brief, we do not need to and cannot (on the basis of (8a) and (8b)) stipulate ‘latest’ and
‘earliest’ in the semantics of the temporal operators until or since. In both cases, the definite
descriptions in (7a) and (7b) refer to the most informative time that satisfies the relevant property
(lt. Bill had been living in Boston until t, in (7a) and lt. Bill has been living in Paris since t, in
(7b). The difference, once again, has to do with the monotonicity of the property.
Finally, we should note that while our account delivers correct interpretations for these temporal
definite descriptions, the Sharvy/Link account would fail quite miserably, since it would require,
for example, that there be a unique day since which Bill has been living in New York for (7b) to
be well-formed. That is simply not the case: if Bill has been living in New York since January
5, 1999, then he has also been living in New York since February 1, 2000, etc. What matters is
that the day when he started living in New York is the most informative, not the unique day falling
under the description.
8 There
are additional arguments in Iatridou (2014).
9 Similarly:
(i) #Miranda’s height is 1.60 but maybe it is 1.65m
10 With
for sure and on the assumption that for sure and maybe have the same modal flavor, (10) is a contradiction.
Without for sure, (10) is presumably an instance of Moore’s Paradox and has whatever defects such instances have.
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Kai von Fintel, Danny Fox, and Sabine Iatridou
5 A potential problem
Imagine a final exam that consists of 20 questions, five of which (q1 q5 ) are designated as
necessary and sufficieng for a passing grade. That is, imagine that if one fails to answer even
one of q1 q5 correctly one fails the test and if one answers all of q1 q5 correctly one passes
the test, with the grade determined by the answers given to the remaining questions (q6 q20 ). If
such is the test, the definite description in (12), it seems to us, would refer to the designated five
questions, technically the sum of q1 q5 (henceforth, just q1 q5 ).
(12) The questions such that if one answers all of them correctly one passes the test11
At first glance, this seems favorable to our analysis. One would be forgiven for thinking
that surely the plurality of q1 q5 is the most informative object here and that this is so even
though there are bigger pluralities of questions that are also such that if one answers all of them
correctly one passes the test, all the way up to the biggest plurality of relevance, q1 q20 . Surely,
if answering q1 q5 correctly lets one pass the test, it follows from that that answering supersets
of those five questions correctly should also let one pass the test.12 So, it would seem, our analysis
predicts that the minimal set of questions that lets one pass the test will be the value of the definite
description in (12).
Unfortunately, things are not that simple. The property that the is applied to in (12) is a
conjunction of the predicate l w.l x.x are questions in w and the predicate l w.l x. in w, if one
answers all of x correctly, one passes the test. Now, the second conjunct is indeed downward
monotone in our sense and thus would give us the right handle on why the minimal set of questions
is the most informative object. But the problem is that the first conjunct is a standard upward
monotone predicate. And thus, the conjunction of the two predicates is, of course, non-monotonic,
which leaves us without an explanation for the observed interpretation of (12).
What we would like to do next is to show how this problem could be resolved with machinery
that has been suggested to account for scope conflicts that arise in intensional contexts (specifically
for a de re interpretation of a quantificational phrase that can be demonstrated to have narrow scope
relative to an intensional operator, so called third readings, see von Fintel and Heim 2011, Chapter
9). Specifically assume that the property in (12) could receive the following rendition, where w0 is
(the world denoted by) a free variable inside the NP questions.
(13) l x.l w. x are questions in w0 and in w, if one answers all of x correctly, one passes the test
To see that this would resolve our problem, take two (plural) individuals X and Y such that both
are questions in w0 . For such individuals the entailment in (13) hold iff the entailment in (14) does:
11 We suspect that definite descriptions of this sort would be more easily formed in languages that are more
comfortable with resumptive pronouns, as suggested by the following Hebrew example:
(i) Ha-Se?elot
Se mi se ?one
naxon
?al kulan
?over et ha-bxina hen q1
The-questions that who that answers correctly on all-them passes acc-the-test are q1
q5 .
q5 .
12 In effect, we are taking for granted that the conditional in (12) is Strawson downward entailing with respect to its
first argument, as argued for in von Fintel 1999. Strictly speaking, we would then suggest that (3b) be modified with
‘entail’ replaced with ‘Strawson entail’.
Definiteness as Maximal Informativeness
173
(14) l w. X are questions in w0 and in w, if one answers all of X correctly, one passes the test
entails
l w. Y are questions in w0 and in w, if one answers all of Y correctly, one passes the test
(15) if one answers all of X correctly, one passes the test
entails
if one answer all of Y correctly, one passes the test
More generally if X and Y are both members of a set A, and B is a function from worlds to sets,
we get the following:
(16) {w : X
{w : X
iff X
iff Y
2 A \ B(w)} ✓ {w : Y 2 A \ B(w)} iff
2 B(w)} ✓ {w : Y 2 B(w)}
Y (when B involves entailment from sets to subsets)
X (when B involves entailment from sets to supersets)
We would thus like to suggest that non-local binding of world variables (inside the NP
questions) is what resolves our problem and explains the meaning of (12). Whether or not this
suggestion is correct is something that we will have to leave for further research.13
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